Fence with minimum cost
A fence is to be built to enclose a rectangular area of 300 square feet. The fence along three sides is to be made of material that costs 3 dollars per foot, and the material for the fourth side costs 12 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.
Shile
1 Answer
Let $x$ and $y$ be the sides oif the rectangular region. Then
\[xy=300 \Rightarrow y=\frac{300}{x}.\]
The cost of the material is
\[C=3(x+y+x)+12 y=6x+15y=6x+15\frac{300}{x}.\]
So
\[C=6x+\frac{4500}{x}.\]
To minimize $C$ we take a derivative to find the critical points
\[C'(x)=6-\frac{4500}{x^2}=0\]
\[\Rightarrow x^2=\frac{4500}{6}=750.\]
Hence
\[x=\sqrt{750} \text{and} y=\frac{300}{\sqrt{750}},\]
are dimensions of the enclosure that is most economical to construct.
Savionf
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